\(\int x^2 (d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x))^n \, dx\) [522]

Optimal result

Integrand size = 28, antiderivative size = 816 \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=-\frac {5 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{128 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {2^{-11-3 n} d^2 e^{-\frac {8 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} 3^{-1-n} d^2 e^{-\frac {6 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-2 (4+n)} d^2 e^{-\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} d^2 e^{-\frac {2 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} d^2 e^{\frac {2 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-2 (4+n)} d^2 e^{\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} 3^{-1-n} d^2 e^{\frac {6 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-11-3 n} d^2 e^{\frac {8 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}} \]

[Out]

-5/128*d^2*(a+b*arcsinh(c*x))^(1+n)*(c^2*d*x^2+d)^(1/2)/b/c^3/(1+n)/(c^2*x^2+1)^(1/2)+2^(-11-3*n)*d^2*(a+b*arc
sinh(c*x))^n*GAMMA(1+n,-8*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^3/exp(8*a/b)/(((-a-b*arcsinh(c*x))/b)^n)
/(c^2*x^2+1)^(1/2)+2^(-7-n)*3^(-1-n)*d^2*(a+b*arcsinh(c*x))^n*GAMMA(1+n,-6*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)
^(1/2)/c^3/exp(6*a/b)/(((-a-b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)+d^2*(a+b*arcsinh(c*x))^n*GAMMA(1+n,-4*(a+b
*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/(2^(8+2*n))/c^3/exp(4*a/b)/(((-a-b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)
-2^(-7-n)*d^2*(a+b*arcsinh(c*x))^n*GAMMA(1+n,-2*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^3/exp(2*a/b)/(((-a
-b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)+2^(-7-n)*d^2*exp(2*a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,2*(a+b*arcsinh
(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^3/(((a+b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)-d^2*exp(4*a/b)*(a+b*arcsinh(c*x
))^n*GAMMA(1+n,4*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/(2^(8+2*n))/c^3/(((a+b*arcsinh(c*x))/b)^n)/(c^2*x^2
+1)^(1/2)-2^(-7-n)*3^(-1-n)*d^2*exp(6*a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,6*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d
)^(1/2)/c^3/(((a+b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)-2^(-11-3*n)*d^2*exp(8*a/b)*(a+b*arcsinh(c*x))^n*GAMMA
(1+n,8*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^3/(((a+b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 816, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5819, 5556, 3388, 2212} \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=\frac {2^{-3 n-11} d^2 e^{-\frac {8 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \Gamma \left (n+1,-\frac {8 (a+b \text {arcsinh}(c x))}{b}\right ) \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}+\frac {2^{-n-7} 3^{-n-1} d^2 e^{-\frac {6 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \Gamma \left (n+1,-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right ) \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}+\frac {2^{-2 (n+4)} d^2 e^{-\frac {4 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \Gamma \left (n+1,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right ) \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}-\frac {2^{-n-7} d^2 e^{-\frac {2 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \Gamma \left (n+1,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right ) \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}-\frac {5 d^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^{n+1}}{128 b c^3 (n+1) \sqrt {c^2 x^2+1}}+\frac {2^{-n-7} d^2 e^{\frac {2 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {2^{-2 (n+4)} d^2 e^{\frac {4 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {2^{-n-7} 3^{-n-1} d^2 e^{\frac {6 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {2^{-3 n-11} d^2 e^{\frac {8 a}{b}} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {c^2 x^2+1}} \]

[In]

Int[x^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n,x]

[Out]

(-5*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^(1 + n))/(128*b*c^3*(1 + n)*Sqrt[1 + c^2*x^2]) + (2^(-11 - 3*
n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-8*(a + b*ArcSinh[c*x]))/b])/(c^3*E^((8*a)/b)*
Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (2^(-7 - n)*3^(-1 - n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSi
nh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcSinh[c*x]))/b])/(c^3*E^((6*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x
])/b))^n) + (d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(
4 + n))*c^3*E^((4*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) - (2^(-7 - n)*d^2*Sqrt[d + c^2*d*x^2]
*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b])/(c^3*E^((2*a)/b)*Sqrt[1 + c^2*x^2]*(-((a +
b*ArcSinh[c*x])/b))^n) + (2^(-7 - n)*d^2*E^((2*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (
2*(a + b*ArcSinh[c*x]))/b])/(c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) - (d^2*E^((4*a)/b)*Sqrt[d + c^2
*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(4 + n))*c^3*Sqrt[1 + c^2*x^2]*
((a + b*ArcSinh[c*x])/b)^n) - (2^(-7 - n)*3^(-1 - n)*d^2*E^((6*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^
n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b])/(c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) - (2^(-11 - 3*n
)*d^2*E^((8*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (8*(a + b*ArcSinh[c*x]))/b])/(c^3*Sq
rt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n)

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5819

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*
c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1),
x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int x^n \cosh ^6\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^3 \sqrt {1+c^2 x^2}} \\ & = \frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (-\frac {5 x^n}{128}+\frac {1}{128} x^n \cosh \left (\frac {8 a}{b}-\frac {8 x}{b}\right )+\frac {1}{32} x^n \cosh \left (\frac {6 a}{b}-\frac {6 x}{b}\right )+\frac {1}{32} x^n \cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )-\frac {1}{32} x^n \cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {5 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{128 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {8 a}{b}-\frac {8 x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{128 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {6 a}{b}-\frac {6 x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b c^3 \sqrt {1+c^2 x^2}}-\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {5 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{128 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {8 i a}{b}-\frac {8 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{256 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {8 i a}{b}-\frac {8 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{256 b c^3 \sqrt {1+c^2 x^2}}-\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^3 \sqrt {1+c^2 x^2}}-\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {6 i a}{b}-\frac {6 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {6 i a}{b}-\frac {6 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {5 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{128 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {2^{-11-3 n} d^2 e^{-\frac {8 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} 3^{-1-n} d^2 e^{-\frac {6 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {4^{-4-n} d^2 e^{-\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} d^2 e^{-\frac {2 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} d^2 e^{\frac {2 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {4^{-4-n} d^2 e^{\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} 3^{-1-n} d^2 e^{\frac {6 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-11-3 n} d^2 e^{\frac {8 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.62 (sec) , antiderivative size = 667, normalized size of antiderivative = 0.82 \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=-\frac {2^{-11-3 n} 3^{-1-n} d^3 e^{-\frac {8 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^{-n} \left (-3^{1+n} b (1+n) \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )-4^{2+n} b e^{\frac {2 a}{b}} (1+n) \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )-2^{3+n} 3^{1+n} b e^{\frac {4 a}{b}} (1+n) \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+3^{1+n} 4^{2+n} b e^{\frac {6 a}{b}} (1+n) \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+e^{\frac {8 a}{b}} \left (5\ 2^{4+3 n} 3^{1+n} a \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^n+5\ 2^{4+3 n} 3^{1+n} b \text {arcsinh}(c x) \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^n-3^{1+n} 4^{2+n} b e^{\frac {2 a}{b}} (1+n) \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+2^{3+n} 3^{1+n} b e^{\frac {4 a}{b}} (1+n) \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+4^{2+n} b e^{\frac {6 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )+4^{2+n} b e^{\frac {6 a}{b}} n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )+3^{1+n} b e^{\frac {8 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )+3^{1+n} b e^{\frac {8 a}{b}} n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{b c^3 (1+n) \sqrt {d+c^2 d x^2}} \]

[In]

Integrate[x^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n,x]

[Out]

-((2^(-11 - 3*n)*3^(-1 - n)*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*(-(3^(1 + n)*b*(1 + n)*(a/b + ArcSinh
[c*x])^n*Gamma[1 + n, (-8*(a + b*ArcSinh[c*x]))/b]) - 4^(2 + n)*b*E^((2*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*G
amma[1 + n, (-6*(a + b*ArcSinh[c*x]))/b] - 2^(3 + n)*3^(1 + n)*b*E^((4*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*Ga
mma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b] + 3^(1 + n)*4^(2 + n)*b*E^((6*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*Gam
ma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b] + E^((8*a)/b)*(5*2^(4 + 3*n)*3^(1 + n)*a*(-((a + b*ArcSinh[c*x])^2/b^2)
)^n + 5*2^(4 + 3*n)*3^(1 + n)*b*ArcSinh[c*x]*(-((a + b*ArcSinh[c*x])^2/b^2))^n - 3^(1 + n)*4^(2 + n)*b*E^((2*a
)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b] + 2^(3 + n)*3^(1 + n)*b*E^
((4*a)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b] + 4^(2 + n)*b*E^((6*a
)/b)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b] + 4^(2 + n)*b*E^((6*a)/b)*n*(-((a
+ b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b] + 3^(1 + n)*b*E^((8*a)/b)*(-((a + b*ArcSinh[c
*x])/b))^n*Gamma[1 + n, (8*(a + b*ArcSinh[c*x]))/b] + 3^(1 + n)*b*E^((8*a)/b)*n*(-((a + b*ArcSinh[c*x])/b))^n*
Gamma[1 + n, (8*(a + b*ArcSinh[c*x]))/b])))/(b*c^3*E^((8*a)/b)*(1 + n)*Sqrt[d + c^2*d*x^2]*(-((a + b*ArcSinh[c
*x])^2/b^2))^n))

Maple [F]

\[\int x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{n}d x\]

[In]

int(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^n,x)

[Out]

int(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^n,x)

Fricas [F]

\[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x^{2} \,d x } \]

[In]

integrate(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^n,x, algorithm="fricas")

[Out]

integral((c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2)*sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)^n, x)

Sympy [F(-1)]

Timed out. \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=\text {Timed out} \]

[In]

integrate(x**2*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**n,x)

[Out]

Timed out

Maxima [F]

\[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x^{2} \,d x } \]

[In]

integrate(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^n,x, algorithm="maxima")

[Out]

integrate((c^2*d*x^2 + d)^(5/2)*(b*arcsinh(c*x) + a)^n*x^2, x)

Giac [F]

\[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x^{2} \,d x } \]

[In]

integrate(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^n,x, algorithm="giac")

[Out]

integrate((c^2*d*x^2 + d)^(5/2)*(b*arcsinh(c*x) + a)^n*x^2, x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \]

[In]

int(x^2*(a + b*asinh(c*x))^n*(d + c^2*d*x^2)^(5/2),x)

[Out]

int(x^2*(a + b*asinh(c*x))^n*(d + c^2*d*x^2)^(5/2), x)